"I shall now show by our figure that the Lorentz hypothesis is
entirely equivalent to the new conception of space and time through
which it may much more readily be understood.

"If, for simplicity's sake we ignore y and z and consider
a world of one space-dimension, then parallel strips --
an upright one like the
t-axis, and one inclined to it --
represent the path respectively of a stationary and a uniformly moving body
which in both cases maintain a constant spatial extent."

What Minkowski means by "constant spatial extent" will be clarified
shortly. What matters for now is that the bodies are not changing
their shape for any reason unrelated to the Lorentz contraction.

"If OA′ [i. e. the t′ axis]
is parallel to the second strip, we can introduce t′ as
time and x′ as the space coordinate, and the second body
then appears [in the primed coordinate-system]
to be at rest, and the first to be in uniform motion.

"We now assume that the first body
conceived as at rest has the length l,
that is, the cross-section PP of the first strip on the x-axis =
l × OC, where OC denotes the unit on the x=axis ;
and on the other hand that the second body conceived as at rest
has the same length l, that is, the cross-section of the second strip,
measured parallel to the x′-axis gives the equation
Q′Q′ = l × OC′."

In other words, we are to use the special "primed meters" in which
the primed-coordinate unit hyperbola is still a unit hyperbola.
It is not clear that Minkowski actually does this in practice.
Like most first attempts at anything, the first use of world-lines
is hesitant and confused; one regrets that Minkowski does not think of
introducing hyperbolic trigonometric-functions and dispensing with the
strange units entirely. Later in the talk he will move in this
direction, to the great relief of the commentator.

"We now have in these two bodies constructions of two equal
Lorentz electrons, one at rest and one in uniform motion."

This is just an aside, relating Minkowski's terminology to Lorentz's.
The details of Lorentz's classical model of the atom are unimportant for
understanding Minkowski's work; the curious reader may find
a clear introductory account here.

"If we keep the original coordinates x, t, fixed, then the section
QQ of the respective strip parallel to the x-axis, must be regarded
as an extension of the second electron."

The translation of this sentence is difficult; Carus's (given here)
does not seem to make sense.
Ausdehnung ("extension") could mean "spreading",
but the segment QQ is
shorter than most others under consideration.
It can also mean "extent". Saha
translates "Now if we stick to our original coordinates,
then the extension of the second electron is given by the
cross section QQ of the strip belonging to it measured
parallel to the x-axis."

"Now it is clear since Q′Q′ = l × OC′
that QQ = l × OD′."

"A simple calculation shows that if (dx/dt) = v for the second strip.

and therefore also

"But this is the meaning of the hypothesis of Lorentz on the contraction
of electrons in motion."

The "simple calculation" which Minkowski omits involves only elementary
mathematics, but is still fairly involved.
First, recall that the point A′ has (unprimed) coordinates
pe + qf, and that the direction of
the x′-axis is given by the unit vector

e′ =
(q/L)e + (p/L)f,

where
L² = p² + q².

The point B′ is where a line through A′ in the
x′-direction intersects the asymptote
ct = x. Let the segment A′B′, which
points in the x′ direction and is therefore
proportional to e′, have
magnitude m. Then

OB′ =
OA′ + A′B′ =
(pe + qf) +
m ((q/L)e + (p/L)f)

Since OB′ is on the asymptote, the x and ct
components must be equal. Therefore

p + mq/L = q + mp/L

that is, m = L.

Now OC′ has the same magnitude as A′B′
and also lies along the x′-axis. Therefore

OC′ = qe + pf

It can be seen from the figure that OC′ is the vector sum of
OD′ and some vector D′C′ in the ct
direction, i.e. parallel to OA′. Let R be the length of OD′,
so that OD′ = Re, and let λ be the ratio
between the magnitudes of OA′ and D′C′.
Then we have:

(R + λp)e + λqf =
qe + pf

Equating the components, we find

R = q - λp
λ = p/q

Solving the two equations simultaneously, we have R = q - p²q.
Bring the right side to a common denominator and multiply by 1 in
the form q/q to find R = q(1 - (p²/q²))

But p/q (that is, λ) is just the velocity of the second body
(divided by c), as Minkowski has already
explained, so we can write

R = q(1 - (v²/c²))

This looks promising. To complete the proof, let h be the magnitude of
OC. From the figure, when x = h, ct is also h, and
the hyperbola is given by c²t² - x² =
h². In particular, since A′ is also on the hyperbola,
q² - p² = h². Dividing by
q² and using v/c = p/q, we find (h/q)² =
1 - (v/c)². Solve this for q² and insert it into
the square of the formula
for R² we have already obtained. Behold:

R² = h²(1 - (v/c)²)

Since R is the magnitude of OD′ and h is the magnitude
of OC, the square-root of this equation gives Minkowski's

Although I was unaware of it when I wrote the above, I have since
learned that Arnold Sommerfeld gives a different proof in the
notes to
Das Relativitätsprinzip: Eine Sammlung von Abhandlungen
[Leipzig: Teubner, 1913; English translation
The Principle of Relativity: A Collection of Original
Memoirs, Dover reprint, no date]. Sommerfeld's argument
uses only trigonometry, and no vectors, so some readers may
prefer it. It is available online in German
here.

"If, on the other hand, adopting the system of reference x′
t′, we regard the second electron as at rest, then the
length of the first will be denoted by the cross section P′P′
of its strip parallel to OC′, and we would find the first
electron shortened in exactly the same proportion with reference to the
second. For it is according to the figure:

P′P′ : Q′Q′ = OD : OC′ =
OD′ : OC = QQ : PP

That is, the Lorentz transformations are symmetric
with regard to velocity; we cannot say which of the two strips is "actually"
moving.

"Lorentz called the combination t′ of x and
t the place-time[Ortszeit] of the uniformly moving electron
and used a physical construction of this conception for the better
understanding of the contraction hypothesis. But it remained for
A. Einstein [Annalen der Physik, XVII, 1905, p. 891;
Jahrbuch der Radioaktivität und Electronik, IV, 1907,
p. 411] to recognize clearly that the time of one electron was
just as good as that of the other, that is, that t and t′
are to be treated alike. Thus time was the first to be discarded
as a concept determined uniquely by phenomena."

"Neither Einstein nor Lorentz disturbed the conception of space, perhaps
for the reason that in the special transformation where the x′
t′ plane coincides with the x, t plane it is possible to
interpret the x-axis of space as remaining fixed in position."

I do not quite understand what Minkowski means by this.

"To loftily ignore the conception of space in similar wise is doubtless due to
the boldness of mathematical discipline!"

"After this further step,
which however is indispensable for a true understanding of the group
G(c), the expression postulate of relativity for the
demand for an invariance in the group G(c), seems to me
very weak. Since the postulate comes to mean that phenomena occur
only in the four-dimensional world of space and time but the projection
into space and time can still be assumed with a certain degree of freedom,
I would rather call this proposition the postulate of the absolute world
(or for short, world-postulate)."

Indeed, the use of the term "relativity" to describe the new world-view
is both a blessing and a curse; many physicists over the next century will
regret that some other term, perhaps "theory of general invariance", had
not been selected. However, it seems probable that such a name
(to say nothing of "theory of the absolute world"!) would have
had much less impact on the general public.
The false association of "relativity" with
moral and social "relativism" will unquestionably be a
major factor in the enormous popular fad for all things Einstein in the
1920s, and will help to ensure that "modern physics"
joins "modern art", "modern music", and all the other fashionable
modernities in the Twentieth Century pantheon.
Of course, this association will then tragically
backfire when new barbarians trumpet it as evidence that "Jewish
science" is part of a nefarious plot to undermine Western values.

In some future blog, I will explore in more detail the
nature of "revolutionary" scientific developments
in the early 1900s, and the ways in which they represent
not a break with tradition, but a natural extension of it.
For now I will simply remark that the Nineteenth was
a revolutionary century, renewing its life through
an organic process of constant upheaval.